Approximate Moving Least-Squares Approximation for Time-Dependent PDEs

For multivariate problems with many scattered data locations the use of radial functions has proven to be advantageous. However, using the usual radial basis function approach one needs to solve a large (possibly dense) linear system. In the moving least squares (MLS) method one obtains a best approximation of the given data in a (moving) weighted least-squares sense. The computational burden is now shifted, and one needs to solve many small linear systems. Recently we have employed the theory of approximate approximations (see [12]) to develop a completely matrix-free approximate MLS approximation algorithm. So far we have only discussed applications of this method to scattered data approximation problems (see [5], [6]). In this paper we present a comparison of two approaches to the solution of time dependent (parabolic) PDEs of the form ∂u ∂t (x, t) = Lu(x, t) + F (x, t), x ∈ Ω ⊂ IR, t > 0, based on the use of approximate moving least-squares approximation. In the first approach one assumes the solution to be an approximate MLS approximation of the form u(x, t) = N ∑ j=1 αj(t)ψj(x), x ∈ IR, where the generating functions ψj(x) = Ψ(‖x − xj‖) satisfy certain moment conditions to ensure a desired approximation order. This leads to a system of ordinary differential equations for the coefficients αj(t). Many traditional techniques can be applied to solve this ODE or DAE system. For the second approach one first discretizes in time, and then applies approximate MLS collocation to the spatial part. This part of the solution is analogous to scattered Hermite interpolation. Similarities and differences of the two methods as well as numerical experiments are presented. Gregory E. Fasshauer